The major roadblock would be the amount time involved to implement the feature, and that it would require the map file format to be changed - which would most likely break compatibility with the many engines that support TMX files.
Yes, it is possible, but it would be a major undertaking.
You can find Tiled's source code here: https://github.com/bjorn/tiled, if you would like to implement the feature in the editor. I'm not quite sure what you're looking for, but you can certainly customize Tiled to suit your needs.
You wouldn't necessarily need a separate object layer for each tile layer; you could have a block of code that loops through each tile layer and for each object that had been loaded, apply the properties to the appropriate region of tiles.
The objects wouldn't have to be rectangles, though they'd be the easiest to write the code for and it would be fine for them to overlap to make more complex shapes.
If you didn't want a region's properties to be applied to a particular layer, you could add a property to the object or object layer that wouldn't be added to tiles, but would tell your block of code to only apply the region's properties to any layer specified by index.